Class equation decomposition

Proposition (Decomposition underlying the class equation). Let $G$ be a finite group . Consider the action of $G$ on itself by conjugation (see conjugation action ). Then:

1. $G$ is a disjoint union of its conjugacy classes.
2. The elements with singleton conjugacy class are exactly the center $Z(G)$.
3. For each $x\in G$, the size of the conjugacy class of $x$ equals the index $[G:C_G(x)]$, where $C_G(x)$ is the centralizer , and this is a consequence of the orbit–stabilizer theorem .

In particular, choosing one representative $x_i$ from each conjugacy class outside the center yields the decomposition

Context. This is the structural content behind the class equation : it turns the conjugation action into a counting identity, a key tool for $p$-groups and Sylow theory.

Proof sketch. (1) Orbits of an action partition the underlying set. (2) A conjugacy class of $x$ has size $1$ iff $gxg^{-1}=x$ for all $g$, i.e. $x\in Z(G)$. (3) Apply orbit–stabilizer to the conjugation action: the stabilizer of $x$ is precisely $C_G(x)$.